Result for Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Specification

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \left(t - x\right) \end{array} \]

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

public static double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

def code(x, y, z, t):
	return x + ((y - z) * (t - x))

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end

function tmp = code(x, y, z, t)
	tmp = x + ((y - z) * (t - x));
end

code[x_, y_, z_, t_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
x + \left(y - z\right) \cdot \left(t - x\right)
\end{array}

Sampling outcomes in binary64 precision:

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \left(t - x\right) \end{array} \]

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

public static double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

def code(x, y, z, t):
	return x + ((y - z) * (t - x))

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end

function tmp = code(x, y, z, t)
	tmp = x + ((y - z) * (t - x));
end

code[x_, y_, z_, t_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
x + \left(y - z\right) \cdot \left(t - x\right)
\end{array}

Alternative 1: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \left(t - x\right) \end{array} \]

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

public static double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

def code(x, y, z, t):
	return x + ((y - z) * (t - x))

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end

function tmp = code(x, y, z, t)
	tmp = x + ((y - z) * (t - x));
end

code[x_, y_, z_, t_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
x + \left(y - z\right) \cdot \left(t - x\right)
\end{array}

Derivation

Initial program 100.0%
\[x + \left(y - z\right) \cdot \left(t - x\right) \]
Add Preprocessing
Add Preprocessing

Alternative 2: 70.5% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+182}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) t)))
   (if (<= z -1.02e+62)
     t_1
     (if (<= z 2.3e+107) (fma (- t x) y x) (if (<= z 7e+182) t_1 (* z x))))))

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * t;
	double tmp;
	if (z <= -1.02e+62) {
		tmp = t_1;
	} else if (z <= 2.3e+107) {
		tmp = fma((t - x), y, x);
	} else if (z <= 7e+182) {
		tmp = t_1;
	} else {
		tmp = z * x;
	}
	return tmp;
}

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * t)
	tmp = 0.0
	if (z <= -1.02e+62)
		tmp = t_1;
	elseif (z <= 2.3e+107)
		tmp = fma(Float64(t - x), y, x);
	elseif (z <= 7e+182)
		tmp = t_1;
	else
		tmp = Float64(z * x);
	end
	return tmp
end

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -1.02e+62], t$95$1, If[LessEqual[z, 2.3e+107], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 7e+182], t$95$1, N[(z * x), $MachinePrecision]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_1 := \left(y - z\right) \cdot t\\
\mathbf{if}\;z \leq -1.02 \cdot 10^{+62}:\\
\;\;\;\;t\_1\\

\mathbf{elif}\;z \leq 2.3 \cdot 10^{+107}:\\
\;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\

\mathbf{elif}\;z \leq 7 \cdot 10^{+182}:\\
\;\;\;\;t\_1\\

\mathbf{else}:\\
\;\;\;\;z \cdot x\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if z < -1.02000000000000002e62 or 2.3e107 < z < 7.00000000000000045e182
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \color{blue}{t \cdot \left(y - z\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \left(y - z\right) \cdot \color{blue}{t} \]
  2. lower-*.f64N/A
    \[\leadsto \left(y - z\right) \cdot \color{blue}{t} \]
  3. lift--.f6464.1
    \[\leadsto \left(y - z\right) \cdot t \]
5. Applied rewrites64.1%
  \[\leadsto \color{blue}{\left(y - z\right) \cdot t} \]
if -1.02000000000000002e62 < z < 2.3e107
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6486.5
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites86.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
if 7.00000000000000045e182 < z
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \color{blue}{x + -1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto -1 \cdot \left(z \cdot \left(t - x\right)\right) + \color{blue}{x} \]
  2. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \left(t - x\right) + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(-1 \cdot z, \color{blue}{t - x}, x\right) \]
  4. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), \color{blue}{t} - x, x\right) \]
  5. lower-neg.f64N/A
    \[\leadsto \mathsf{fma}\left(-z, \color{blue}{t} - x, x\right) \]
  6. lift--.f6492.4
    \[\leadsto \mathsf{fma}\left(-z, t - \color{blue}{x}, x\right) \]
5. Applied rewrites92.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(-z, t - x, x\right)} \]
6. Taylor expanded in t around 0
  \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto x \cdot z + x \]
  2. lower-fma.f6458.8
    \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
8. Applied rewrites58.8%
  \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
9. Taylor expanded in z around inf
  \[\leadsto x \cdot z \]
10. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto z \cdot x \]
  2. lower-*.f6458.8
    \[\leadsto z \cdot x \]
11. Applied rewrites58.8%
  \[\leadsto z \cdot x \]
Recombined 3 regimes into one program.
Add Preprocessing

Alternative 3: 50.6% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+217}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq -1700000000000 \lor \neg \left(y \leq 26\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.8e+217)
   (* t y)
   (if (or (<= y -1700000000000.0) (not (<= y 26.0)))
     (* (- x) y)
     (fma x z x))))

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.8e+217) {
		tmp = t * y;
	} else if ((y <= -1700000000000.0) || !(y <= 26.0)) {
		tmp = -x * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.8e+217)
		tmp = Float64(t * y);
	elseif ((y <= -1700000000000.0) || !(y <= 26.0))
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

code[x_, y_, z_, t_] := If[LessEqual[y, -1.8e+217], N[(t * y), $MachinePrecision], If[Or[LessEqual[y, -1700000000000.0], N[Not[LessEqual[y, 26.0]], $MachinePrecision]], N[((-x) * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;y \leq -1.8 \cdot 10^{+217}:\\
\;\;\;\;t \cdot y\\

\mathbf{elif}\;y \leq -1700000000000 \lor \neg \left(y \leq 26\right):\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, z, x\right)\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if y < -1.8000000000000001e217
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6494.4
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites94.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
6. Taylor expanded in x around 0
  \[\leadsto t \cdot \color{blue}{y} \]
7. Step-by-step derivation
  1. lower-*.f6473.6
    \[\leadsto t \cdot y \]
8. Applied rewrites73.6%
  \[\leadsto t \cdot \color{blue}{y} \]
if -1.8000000000000001e217 < y < -1.7e12 or 26 < y
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around inf
  \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot \color{blue}{y} \]
  2. lower-*.f64N/A
    \[\leadsto \left(t - x\right) \cdot \color{blue}{y} \]
  3. lift--.f6480.0
    \[\leadsto \left(t - x\right) \cdot y \]
5. Applied rewrites80.0%
  \[\leadsto \color{blue}{\left(t - x\right) \cdot y} \]
6. Taylor expanded in x around inf
  \[\leadsto \left(-1 \cdot x\right) \cdot y \]
7. Step-by-step derivation
  1. mul-1-negN/A
    \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]
  2. lift-neg.f6449.1
    \[\leadsto \left(-x\right) \cdot y \]
8. Applied rewrites49.1%
  \[\leadsto \left(-x\right) \cdot y \]
if -1.7e12 < y < 26
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \color{blue}{x + -1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto -1 \cdot \left(z \cdot \left(t - x\right)\right) + \color{blue}{x} \]
  2. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \left(t - x\right) + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(-1 \cdot z, \color{blue}{t - x}, x\right) \]
  4. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), \color{blue}{t} - x, x\right) \]
  5. lower-neg.f64N/A
    \[\leadsto \mathsf{fma}\left(-z, \color{blue}{t} - x, x\right) \]
  6. lift--.f6488.4
    \[\leadsto \mathsf{fma}\left(-z, t - \color{blue}{x}, x\right) \]
5. Applied rewrites88.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(-z, t - x, x\right)} \]
6. Taylor expanded in t around 0
  \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto x \cdot z + x \]
  2. lower-fma.f6456.2
    \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
8. Applied rewrites56.2%
  \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
Recombined 3 regimes into one program.
Final simplification54.3%
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+217}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq -1700000000000 \lor \neg \left(y \leq 26\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
Add Preprocessing

Alternative 4: 84.9% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 100\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, t - x, x\right)\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -0.5) (not (<= y 100.0)))
   (fma (- t x) y x)
   (fma (- z) (- t x) x)))

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -0.5) || !(y <= 100.0)) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = fma(-z, (t - x), x);
	}
	return tmp;
}

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -0.5) || !(y <= 100.0))
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = fma(Float64(-z), Float64(t - x), x);
	end
	return tmp
end

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -0.5], N[Not[LessEqual[y, 100.0]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], N[((-z) * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 100\right):\\
\;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-z, t - x, x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if y < -0.5 or 100 < y
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6482.3
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites82.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
if -0.5 < y < 100
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \color{blue}{x + -1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto -1 \cdot \left(z \cdot \left(t - x\right)\right) + \color{blue}{x} \]
  2. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \left(t - x\right) + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(-1 \cdot z, \color{blue}{t - x}, x\right) \]
  4. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), \color{blue}{t} - x, x\right) \]
  5. lower-neg.f64N/A
    \[\leadsto \mathsf{fma}\left(-z, \color{blue}{t} - x, x\right) \]
  6. lift--.f6491.5
    \[\leadsto \mathsf{fma}\left(-z, t - \color{blue}{x}, x\right) \]
5. Applied rewrites91.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(-z, t - x, x\right)} \]
Recombined 2 regimes into one program.
Final simplification86.5%
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.5 \lor \neg \left(y \leq 100\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-z, t - x, x\right)\\ \end{array} \]
Add Preprocessing

Alternative 5: 38.1% accurate, 0.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+44}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-276}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.16e+44)
   (* z x)
   (if (<= z 2.5e-276) (* t y) (if (<= z 3.2e-21) x (* z x)))))

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.16e+44) {
		tmp = z * x;
	} else if (z <= 2.5e-276) {
		tmp = t * y;
	} else if (z <= 3.2e-21) {
		tmp = x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.16d+44)) then
        tmp = z * x
    else if (z <= 2.5d-276) then
        tmp = t * y
    else if (z <= 3.2d-21) then
        tmp = x
    else
        tmp = z * x
    end if
    code = tmp
end function

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.16e+44) {
		tmp = z * x;
	} else if (z <= 2.5e-276) {
		tmp = t * y;
	} else if (z <= 3.2e-21) {
		tmp = x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

def code(x, y, z, t):
	tmp = 0
	if z <= -1.16e+44:
		tmp = z * x
	elif z <= 2.5e-276:
		tmp = t * y
	elif z <= 3.2e-21:
		tmp = x
	else:
		tmp = z * x
	return tmp

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.16e+44)
		tmp = Float64(z * x);
	elseif (z <= 2.5e-276)
		tmp = Float64(t * y);
	elseif (z <= 3.2e-21)
		tmp = x;
	else
		tmp = Float64(z * x);
	end
	return tmp
end

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.16e+44)
		tmp = z * x;
	elseif (z <= 2.5e-276)
		tmp = t * y;
	elseif (z <= 3.2e-21)
		tmp = x;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

code[x_, y_, z_, t_] := If[LessEqual[z, -1.16e+44], N[(z * x), $MachinePrecision], If[LessEqual[z, 2.5e-276], N[(t * y), $MachinePrecision], If[LessEqual[z, 3.2e-21], x, N[(z * x), $MachinePrecision]]]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;z \leq -1.16 \cdot 10^{+44}:\\
\;\;\;\;z \cdot x\\

\mathbf{elif}\;z \leq 2.5 \cdot 10^{-276}:\\
\;\;\;\;t \cdot y\\

\mathbf{elif}\;z \leq 3.2 \cdot 10^{-21}:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;z \cdot x\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if z < -1.1600000000000001e44 or 3.2000000000000002e-21 < z
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \color{blue}{x + -1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto -1 \cdot \left(z \cdot \left(t - x\right)\right) + \color{blue}{x} \]
  2. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \left(t - x\right) + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(-1 \cdot z, \color{blue}{t - x}, x\right) \]
  4. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), \color{blue}{t} - x, x\right) \]
  5. lower-neg.f64N/A
    \[\leadsto \mathsf{fma}\left(-z, \color{blue}{t} - x, x\right) \]
  6. lift--.f6477.3
    \[\leadsto \mathsf{fma}\left(-z, t - \color{blue}{x}, x\right) \]
5. Applied rewrites77.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(-z, t - x, x\right)} \]
6. Taylor expanded in t around 0
  \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto x \cdot z + x \]
  2. lower-fma.f6440.1
    \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
8. Applied rewrites40.1%
  \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
9. Taylor expanded in z around inf
  \[\leadsto x \cdot z \]
10. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto z \cdot x \]
  2. lower-*.f6439.6
    \[\leadsto z \cdot x \]
11. Applied rewrites39.6%
  \[\leadsto z \cdot x \]
if -1.1600000000000001e44 < z < 2.49999999999999984e-276
1. Initial program 99.9%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6490.3
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites90.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
6. Taylor expanded in x around 0
  \[\leadsto t \cdot \color{blue}{y} \]
7. Step-by-step derivation
  1. lower-*.f6441.8
    \[\leadsto t \cdot y \]
8. Applied rewrites41.8%
  \[\leadsto t \cdot \color{blue}{y} \]
if 2.49999999999999984e-276 < z < 3.2000000000000002e-21
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6491.5
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites91.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto x \]
7. Step-by-step derivation
8. Applied rewrites44.6%
  \[\leadsto x \]
Recombined 3 regimes into one program.
Add Preprocessing

Alternative 6: 83.4% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+61} \lor \neg \left(z \leq 2.9 \cdot 10^{+75}\right):\\ \;\;\;\;\left(-z\right) \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9.8e+61) (not (<= z 2.9e+75)))
   (* (- z) (- t x))
   (fma (- t x) y x)))

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.8e+61) || !(z <= 2.9e+75)) {
		tmp = -z * (t - x);
	} else {
		tmp = fma((t - x), y, x);
	}
	return tmp;
}

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -9.8e+61) || !(z <= 2.9e+75))
		tmp = Float64(Float64(-z) * Float64(t - x));
	else
		tmp = fma(Float64(t - x), y, x);
	end
	return tmp
end

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -9.8e+61], N[Not[LessEqual[z, 2.9e+75]], $MachinePrecision]], N[((-z) * N[(t - x), $MachinePrecision]), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;z \leq -9.8 \cdot 10^{+61} \lor \neg \left(z \leq 2.9 \cdot 10^{+75}\right):\\
\;\;\;\;\left(-z\right) \cdot \left(t - x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if z < -9.8000000000000005e61 or 2.8999999999999998e75 < z
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around inf
  \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \color{blue}{\left(t - x\right)} \]
  2. lower-*.f64N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \color{blue}{\left(t - x\right)} \]
  3. mul-1-negN/A
    \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \left(\color{blue}{t} - x\right) \]
  4. lower-neg.f64N/A
    \[\leadsto \left(-z\right) \cdot \left(\color{blue}{t} - x\right) \]
  5. lift--.f6482.1
    \[\leadsto \left(-z\right) \cdot \left(t - \color{blue}{x}\right) \]
5. Applied rewrites82.1%
  \[\leadsto \color{blue}{\left(-z\right) \cdot \left(t - x\right)} \]
if -9.8000000000000005e61 < z < 2.8999999999999998e75
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6488.5
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites88.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
Recombined 2 regimes into one program.
Final simplification86.0%
\[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+61} \lor \neg \left(z \leq 2.9 \cdot 10^{+75}\right):\\ \;\;\;\;\left(-z\right) \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \]
Add Preprocessing

Alternative 7: 74.0% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-37} \lor \neg \left(t \leq 1.1 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.15e-37) (not (<= t 1.1e-23)))
   (fma (- y z) t x)
   (fma (- t x) y x)))

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.15e-37) || !(t <= 1.1e-23)) {
		tmp = fma((y - z), t, x);
	} else {
		tmp = fma((t - x), y, x);
	}
	return tmp;
}

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.15e-37) || !(t <= 1.1e-23))
		tmp = fma(Float64(y - z), t, x);
	else
		tmp = fma(Float64(t - x), y, x);
	end
	return tmp
end

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.15e-37], N[Not[LessEqual[t, 1.1e-23]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;t \leq -3.15 \cdot 10^{-37} \lor \neg \left(t \leq 1.1 \cdot 10^{-23}\right):\\
\;\;\;\;\mathsf{fma}\left(y - z, t, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if t < -3.15000000000000011e-37 or 1.1e-23 < t
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto x + \left(y - z\right) \cdot \color{blue}{t} \]
4. Step-by-step derivation
5. Applied rewrites85.0%
  \[\leadsto x + \left(y - z\right) \cdot \color{blue}{t} \]
6. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{x + \left(y - z\right) \cdot t} \]
  2. +-commutativeN/A
    \[\leadsto \color{blue}{\left(y - z\right) \cdot t + x} \]
  3. lift--.f64N/A
    \[\leadsto \color{blue}{\left(y - z\right)} \cdot t + x \]
  4. lift-*.f64N/A
    \[\leadsto \color{blue}{\left(y - z\right) \cdot t} + x \]
  5. lower-fma.f64N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, t, x\right)} \]
  6. lift--.f6485.0
    \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, t, x\right) \]
7. Applied rewrites85.0%
  \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, t, x\right)} \]
if -3.15000000000000011e-37 < t < 1.1e-23
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6475.7
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites75.7%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
Recombined 2 regimes into one program.
Final simplification80.8%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-37} \lor \neg \left(t \leq 1.1 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \]
Add Preprocessing

Alternative 8: 63.3% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-37} \lor \neg \left(t \leq 9.8 \cdot 10^{-45}\right):\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, y, x\right)\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.15e-37) (not (<= t 9.8e-45))) (* (- y z) t) (fma (- x) y x)))

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.15e-37) || !(t <= 9.8e-45)) {
		tmp = (y - z) * t;
	} else {
		tmp = fma(-x, y, x);
	}
	return tmp;
}

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.15e-37) || !(t <= 9.8e-45))
		tmp = Float64(Float64(y - z) * t);
	else
		tmp = fma(Float64(-x), y, x);
	end
	return tmp
end

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.15e-37], N[Not[LessEqual[t, 9.8e-45]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision], N[((-x) * y + x), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;t \leq -3.15 \cdot 10^{-37} \lor \neg \left(t \leq 9.8 \cdot 10^{-45}\right):\\
\;\;\;\;\left(y - z\right) \cdot t\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-x, y, x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if t < -3.15000000000000011e-37 or 9.7999999999999996e-45 < t
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in x around 0
  \[\leadsto \color{blue}{t \cdot \left(y - z\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \left(y - z\right) \cdot \color{blue}{t} \]
  2. lower-*.f64N/A
    \[\leadsto \left(y - z\right) \cdot \color{blue}{t} \]
  3. lift--.f6474.2
    \[\leadsto \left(y - z\right) \cdot t \]
5. Applied rewrites74.2%
  \[\leadsto \color{blue}{\left(y - z\right) \cdot t} \]
if -3.15000000000000011e-37 < t < 9.7999999999999996e-45
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6475.5
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites75.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
6. Taylor expanded in x around inf
  \[\leadsto \mathsf{fma}\left(-1 \cdot x, y, x\right) \]
7. Step-by-step derivation
  1. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(x\right), y, x\right) \]
  2. lift-neg.f6469.6
    \[\leadsto \mathsf{fma}\left(-x, y, x\right) \]
8. Applied rewrites69.6%
  \[\leadsto \mathsf{fma}\left(-x, y, x\right) \]
Recombined 2 regimes into one program.
Final simplification72.2%
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-37} \lor \neg \left(t \leq 9.8 \cdot 10^{-45}\right):\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, y, x\right)\\ \end{array} \]
Add Preprocessing

Alternative 9: 68.6% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-21} \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.4e-21) (not (<= y 1.1))) (* (- t x) y) (fma x z x)))

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.4e-21) || !(y <= 1.1)) {
		tmp = (t - x) * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.4e-21) || !(y <= 1.1))
		tmp = Float64(Float64(t - x) * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.4e-21], N[Not[LessEqual[y, 1.1]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;y \leq -2.4 \cdot 10^{-21} \lor \neg \left(y \leq 1.1\right):\\
\;\;\;\;\left(t - x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, z, x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if y < -2.3999999999999999e-21 or 1.1000000000000001 < y
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around inf
  \[\leadsto \color{blue}{y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot \color{blue}{y} \]
  2. lower-*.f64N/A
    \[\leadsto \left(t - x\right) \cdot \color{blue}{y} \]
  3. lift--.f6478.7
    \[\leadsto \left(t - x\right) \cdot y \]
5. Applied rewrites78.7%
  \[\leadsto \color{blue}{\left(t - x\right) \cdot y} \]
if -2.3999999999999999e-21 < y < 1.1000000000000001
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \color{blue}{x + -1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto -1 \cdot \left(z \cdot \left(t - x\right)\right) + \color{blue}{x} \]
  2. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \left(t - x\right) + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(-1 \cdot z, \color{blue}{t - x}, x\right) \]
  4. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), \color{blue}{t} - x, x\right) \]
  5. lower-neg.f64N/A
    \[\leadsto \mathsf{fma}\left(-z, \color{blue}{t} - x, x\right) \]
  6. lift--.f6492.8
    \[\leadsto \mathsf{fma}\left(-z, t - \color{blue}{x}, x\right) \]
5. Applied rewrites92.8%
  \[\leadsto \color{blue}{\mathsf{fma}\left(-z, t - x, x\right)} \]
6. Taylor expanded in t around 0
  \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto x \cdot z + x \]
  2. lower-fma.f6459.9
    \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
8. Applied rewrites59.9%
  \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
Recombined 2 regimes into one program.
Final simplification70.4%
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-21} \lor \neg \left(y \leq 1.1\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
Add Preprocessing

Alternative 10: 48.5% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+123} \lor \neg \left(y \leq 5.5 \cdot 10^{+79}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.7e+123) (not (<= y 5.5e+79))) (* t y) (fma x z x)))

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.7e+123) || !(y <= 5.5e+79)) {
		tmp = t * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.7e+123) || !(y <= 5.5e+79))
		tmp = Float64(t * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.7e+123], N[Not[LessEqual[y, 5.5e+79]], $MachinePrecision]], N[(t * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;y \leq -1.7 \cdot 10^{+123} \lor \neg \left(y \leq 5.5 \cdot 10^{+79}\right):\\
\;\;\;\;t \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, z, x\right)\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if y < -1.70000000000000001e123 or 5.50000000000000007e79 < y
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6488.8
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites88.8%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
6. Taylor expanded in x around 0
  \[\leadsto t \cdot \color{blue}{y} \]
7. Step-by-step derivation
  1. lower-*.f6448.9
    \[\leadsto t \cdot y \]
8. Applied rewrites48.9%
  \[\leadsto t \cdot \color{blue}{y} \]
if -1.70000000000000001e123 < y < 5.50000000000000007e79
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \color{blue}{x + -1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto -1 \cdot \left(z \cdot \left(t - x\right)\right) + \color{blue}{x} \]
  2. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \left(t - x\right) + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(-1 \cdot z, \color{blue}{t - x}, x\right) \]
  4. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), \color{blue}{t} - x, x\right) \]
  5. lower-neg.f64N/A
    \[\leadsto \mathsf{fma}\left(-z, \color{blue}{t} - x, x\right) \]
  6. lift--.f6478.4
    \[\leadsto \mathsf{fma}\left(-z, t - \color{blue}{x}, x\right) \]
5. Applied rewrites78.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(-z, t - x, x\right)} \]
6. Taylor expanded in t around 0
  \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto x \cdot z + x \]
  2. lower-fma.f6449.6
    \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
8. Applied rewrites49.6%
  \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
Recombined 2 regimes into one program.
Final simplification49.4%
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+123} \lor \neg \left(y \leq 5.5 \cdot 10^{+79}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
Add Preprocessing

Alternative 11: 36.7% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{-20} \lor \neg \left(z \leq 3.2 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9.8e-20) (not (<= z 3.2e-21))) (* z x) x))

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.8e-20) || !(z <= 3.2e-21)) {
		tmp = z * x;
	} else {
		tmp = x;
	}
	return tmp;
}

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-9.8d-20)) .or. (.not. (z <= 3.2d-21))) then
        tmp = z * x
    else
        tmp = x
    end if
    code = tmp
end function

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.8e-20) || !(z <= 3.2e-21)) {
		tmp = z * x;
	} else {
		tmp = x;
	}
	return tmp;
}

def code(x, y, z, t):
	tmp = 0
	if (z <= -9.8e-20) or not (z <= 3.2e-21):
		tmp = z * x
	else:
		tmp = x
	return tmp

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -9.8e-20) || !(z <= 3.2e-21))
		tmp = Float64(z * x);
	else
		tmp = x;
	end
	return tmp
end

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -9.8e-20) || ~((z <= 3.2e-21)))
		tmp = z * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -9.8e-20], N[Not[LessEqual[z, 3.2e-21]], $MachinePrecision]], N[(z * x), $MachinePrecision], x]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;z \leq -9.8 \cdot 10^{-20} \lor \neg \left(z \leq 3.2 \cdot 10^{-21}\right):\\
\;\;\;\;z \cdot x\\

\mathbf{else}:\\
\;\;\;\;x\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if z < -9.8000000000000003e-20 or 3.2000000000000002e-21 < z
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in y around 0
  \[\leadsto \color{blue}{x + -1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto -1 \cdot \left(z \cdot \left(t - x\right)\right) + \color{blue}{x} \]
  2. associate-*r*N/A
    \[\leadsto \left(-1 \cdot z\right) \cdot \left(t - x\right) + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(-1 \cdot z, \color{blue}{t - x}, x\right) \]
  4. mul-1-negN/A
    \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(z\right), \color{blue}{t} - x, x\right) \]
  5. lower-neg.f64N/A
    \[\leadsto \mathsf{fma}\left(-z, \color{blue}{t} - x, x\right) \]
  6. lift--.f6473.0
    \[\leadsto \mathsf{fma}\left(-z, t - \color{blue}{x}, x\right) \]
5. Applied rewrites73.0%
  \[\leadsto \color{blue}{\mathsf{fma}\left(-z, t - x, x\right)} \]
6. Taylor expanded in t around 0
  \[\leadsto x + \color{blue}{x \cdot z} \]
7. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto x \cdot z + x \]
  2. lower-fma.f6437.1
    \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
8. Applied rewrites37.1%
  \[\leadsto \mathsf{fma}\left(x, \color{blue}{z}, x\right) \]
9. Taylor expanded in z around inf
  \[\leadsto x \cdot z \]
10. Step-by-step derivation
  1. *-commutativeN/A
    \[\leadsto z \cdot x \]
  2. lower-*.f6436.6
    \[\leadsto z \cdot x \]
11. Applied rewrites36.6%
  \[\leadsto z \cdot x \]
if -9.8000000000000003e-20 < z < 3.2000000000000002e-21
1. Initial program 100.0%
  \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Taylor expanded in z around 0
  \[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
  2. *-commutativeN/A
    \[\leadsto \left(t - x\right) \cdot y + x \]
  3. lower-fma.f64N/A
    \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
  4. lift--.f6492.9
    \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
5. Applied rewrites92.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
6. Taylor expanded in y around 0
  \[\leadsto x \]
7. Step-by-step derivation
8. Applied rewrites33.8%
  \[\leadsto x \]
Recombined 2 regimes into one program.
Final simplification35.2%
\[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{-20} \lor \neg \left(z \leq 3.2 \cdot 10^{-21}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Add Preprocessing

Alternative 12: 18.3% accurate, 15.0× speedup?

\[\begin{array}{l} \\ x \end{array} \]

(FPCore (x y z t) :precision binary64 x)

double code(double x, double y, double z, double t) {
	return x;
}

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

public static double code(double x, double y, double z, double t) {
	return x;
}

def code(x, y, z, t):
	return x

function code(x, y, z, t)
	return x
end

function tmp = code(x, y, z, t)
	tmp = x;
end

code[x_, y_, z_, t_] := x

\begin{array}{l}

\\
x
\end{array}

Derivation

Initial program 100.0%
\[x + \left(y - z\right) \cdot \left(t - x\right) \]
Add Preprocessing
Taylor expanded in z around 0
\[\leadsto \color{blue}{x + y \cdot \left(t - x\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto y \cdot \left(t - x\right) + \color{blue}{x} \]
2. *-commutativeN/A
  \[\leadsto \left(t - x\right) \cdot y + x \]
3. lower-fma.f64N/A
  \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{y}, x\right) \]
4. lift--.f6464.6
  \[\leadsto \mathsf{fma}\left(t - x, y, x\right) \]
Applied rewrites64.6%
\[\leadsto \color{blue}{\mathsf{fma}\left(t - x, y, x\right)} \]
Taylor expanded in y around 0
\[\leadsto x \]
Step-by-step derivation
Applied rewrites18.4%
\[\leadsto x \]
Add Preprocessing

Developer Target 1: 96.7% accurate, 0.6× speedup?

\[\begin{array}{l} \\ x + \left(t \cdot \left(y - z\right) + \left(-x\right) \cdot \left(y - z\right)\right) \end{array} \]

(FPCore (x y z t)
 :precision binary64
 (+ x (+ (* t (- y z)) (* (- x) (- y z)))))

double code(double x, double y, double z, double t) {
	return x + ((t * (y - z)) + (-x * (y - z)));
}

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((t * (y - z)) + (-x * (y - z)))
end function

public static double code(double x, double y, double z, double t) {
	return x + ((t * (y - z)) + (-x * (y - z)));
}

def code(x, y, z, t):
	return x + ((t * (y - z)) + (-x * (y - z)))

function code(x, y, z, t)
	return Float64(x + Float64(Float64(t * Float64(y - z)) + Float64(Float64(-x) * Float64(y - z))))
end

function tmp = code(x, y, z, t)
	tmp = x + ((t * (y - z)) + (-x * (y - z)));
end

code[x_, y_, z_, t_] := N[(x + N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] + N[((-x) * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
x + \left(t \cdot \left(y - z\right) + \left(-x\right) \cdot \left(y - z\right)\right)
\end{array}

34.5% of points produce a very large (infinite) output. You may want to add a precondition. (more)

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 100.0% accurate, 1.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Alternative 1: 100.0% accurate, 1.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Alternative 2: 70.5% accurate, 0.6× speedupMathFPCoreCJuliaWolframTeX?

if z < -1.02000000000000002e62 or 2.3e107 < z < 7.00000000000000045e182

if -1.02000000000000002e62 < z < 2.3e107

if 7.00000000000000045e182 < z

Alternative 3: 50.6% accurate, 0.6× speedupMathFPCoreCJuliaWolframTeX?

if y < -1.8000000000000001e217

if -1.8000000000000001e217 < y < -1.7e12 or 26 < y

if -1.7e12 < y < 26

Alternative 4: 84.9% accurate, 0.6× speedupMathFPCoreCJuliaWolframTeX?

if y < -0.5 or 100 < y

if -0.5 < y < 100

Alternative 5: 38.1% accurate, 0.6× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if z < -1.1600000000000001e44 or 3.2000000000000002e-21 < z

if -1.1600000000000001e44 < z < 2.49999999999999984e-276

if 2.49999999999999984e-276 < z < 3.2000000000000002e-21

Alternative 6: 83.4% accurate, 0.7× speedupMathFPCoreCJuliaWolframTeX?

if z < -9.8000000000000005e61 or 2.8999999999999998e75 < z

if -9.8000000000000005e61 < z < 2.8999999999999998e75

Alternative 7: 74.0% accurate, 0.7× speedupMathFPCoreCJuliaWolframTeX?

if t < -3.15000000000000011e-37 or 1.1e-23 < t

if -3.15000000000000011e-37 < t < 1.1e-23

Alternative 8: 63.3% accurate, 0.7× speedupMathFPCoreCJuliaWolframTeX?

if t < -3.15000000000000011e-37 or 9.7999999999999996e-45 < t

if -3.15000000000000011e-37 < t < 9.7999999999999996e-45

Alternative 9: 68.6% accurate, 0.7× speedupMathFPCoreCJuliaWolframTeX?

if y < -2.3999999999999999e-21 or 1.1000000000000001 < y

if -2.3999999999999999e-21 < y < 1.1000000000000001

Alternative 10: 48.5% accurate, 0.8× speedupMathFPCoreCJuliaWolframTeX?

if y < -1.70000000000000001e123 or 5.50000000000000007e79 < y

if -1.70000000000000001e123 < y < 5.50000000000000007e79

Alternative 11: 36.7% accurate, 0.8× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if z < -9.8000000000000003e-20 or 3.2000000000000002e-21 < z

if -9.8000000000000003e-20 < z < 3.2000000000000002e-21

Alternative 12: 18.3% accurate, 15.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Developer Target 1: 96.7% accurate, 0.6× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Reproduce

Initial Program: 100.0% accurate, 1.0× speedup?

Alternative 1: 100.0% accurate, 1.0× speedup?

Alternative 2: 70.5% accurate, 0.6× speedup?

`if z < -1.02000000000000002e62 or 2.3e107 < z < 7.00000000000000045e182`

`if -1.02000000000000002e62 < z < 2.3e107`

`if 7.00000000000000045e182 < z`

Alternative 3: 50.6% accurate, 0.6× speedup?

`if y < -1.8000000000000001e217`

`if -1.8000000000000001e217 < y < -1.7e12 or 26 < y`

`if -1.7e12 < y < 26`

Alternative 4: 84.9% accurate, 0.6× speedup?

`if y < -0.5 or 100 < y`

`if -0.5 < y < 100`

Alternative 5: 38.1% accurate, 0.6× speedup?

`if z < -1.1600000000000001e44 or 3.2000000000000002e-21 < z`

`if -1.1600000000000001e44 < z < 2.49999999999999984e-276`

`if 2.49999999999999984e-276 < z < 3.2000000000000002e-21`

Alternative 6: 83.4% accurate, 0.7× speedup?

`if z < -9.8000000000000005e61 or 2.8999999999999998e75 < z`

`if -9.8000000000000005e61 < z < 2.8999999999999998e75`

Alternative 7: 74.0% accurate, 0.7× speedup?

`if t < -3.15000000000000011e-37 or 1.1e-23 < t`

`if -3.15000000000000011e-37 < t < 1.1e-23`

Alternative 8: 63.3% accurate, 0.7× speedup?

`if t < -3.15000000000000011e-37 or 9.7999999999999996e-45 < t`

`if -3.15000000000000011e-37 < t < 9.7999999999999996e-45`

Alternative 9: 68.6% accurate, 0.7× speedup?

`if y < -2.3999999999999999e-21 or 1.1000000000000001 < y`

`if -2.3999999999999999e-21 < y < 1.1000000000000001`

Alternative 10: 48.5% accurate, 0.8× speedup?

`if y < -1.70000000000000001e123 or 5.50000000000000007e79 < y`

`if -1.70000000000000001e123 < y < 5.50000000000000007e79`

Alternative 11: 36.7% accurate, 0.8× speedup?

`if z < -9.8000000000000003e-20 or 3.2000000000000002e-21 < z`

`if -9.8000000000000003e-20 < z < 3.2000000000000002e-21`

Alternative 12: 18.3% accurate, 15.0× speedup?

Developer Target 1: 96.7% accurate, 0.6× speedup?